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The
figure to the right depicts some of the possible strings that
can be composed from this vocabulary. Note that the number of
elements of the vocabulary is 3. The various ellipses depict
all of the possible strings formed from this vocabulary of length
0 through 10. The number of strings in each of these sets is
also shown. This value is simply 3, the number of elements in
the vocabulary, raised to the power equal to the length of the
string. For example, there are 3 to the 9 or 19,683 strings of
length 9.
The
continuation dots that appear below the ellipse of length 10
is intended to remind us that we could continue to generate these
sets for any length n as long as n is finite. The capital Greek
letter Sigma followed by an * is usually used to refer to the
set of all sets of strings up to length n. This is read as Sigma
star. The large ellipse is intended to represent this set in
the figure to the right. Only the sets 0 to 10 are explicitly
represented. Note that they are each subsets of Sigma star.
Clicking
on any of these subsets (except for Sigma 10) will take you to
a page that explicitly lists all of the strings that are elements
of this set. Note that Sigma 9 contains 19,683 such strings and
if you click on it it will take quite a while to load. These
sets have been explicitly made available to you and the number
of elements in each of these subsets computed so that you can
begin to appreciate the size of unconstrained combinatoric spaces
even when the vocabulary or elements of the system are small.
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Consider
this construction in light of the levels hypothesis. Consider
this unconstrained space as the physical level. That is there
is no physical law that precludes writing down any sequence from
this vocabulary of any arbitrary length. However, the game of
Tic-Tac-Toe is only realized on strings that are exactly of length
9. This subset is shown in green in the figure above and is still
quite large. But note that this is a "law" of Tic-Tac-Toe
at the "symbol level," and not a law that holds at
the "physical level." A subset of this set of strings
of length 9 is represented by the blue ellipse. This subset has
only 509 strings. This reduction was achieved by using additional
"laws" of Tic-Tac-Toe to compute this subset. This
subset contains only strings that have either the same number
of 'Xs' and 'Os'; or exactly one more 'X' than the number
of 'Os'. These laws reflect that X always begins and the players
must alternate their moves.
But
a Tic-Tac-Toe game is a sequence of these strings where the sequence
represents a particular sequence of moves. The next figure on
the right helps to illustrate this idea. A sequence can be obtained
by taking the cross-products of sigma star with itself. For example,
the 4-tuple of strings (X#X, X###, O, XX) is an element that
is obtained by a crossing sigma star with itself 4 times.
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The
resulting spaces of possible sequences is, of course, really
enormous. Illustrated to the right are subsets that are obtained
by crossing sigma 9 with itself up to 9 times. Syntactically
correct Tic-Tac-Toe games are a subset of the subset represented
in green. Note that this set is itself enormous. It contains
19,863 to the 9 elements or 443,426,488,243,037,806,754,782,544,259,268,477,000
elements. If we consider only the 509 legal elements, this cross
product is equal to 509 to the 9 which is still a rather large
number of elements; namely 2,293,295,617,071,746,258,042,880
elements.
However,
if we take into account the laws of Tic-Tac-Toe that govern the
transitions from one state to the next, then the number of possible
sequences is less than 9! or 362,880.
These
enormous reductions in the sequences that are actually observed
can not be explained by "physical laws", but they are
explained by assuming that there are laws that govern the syntax
or computation of these sequences.
Finally,
note that laws also govern a semantic level. One such law is:
(a) make a winning move whenever and as soon as possible. Another
is that if (a) is not possible, then block any imminent winning
move by your opponent whenever possible. Note, nothing in the
syntax of the game requires or leads to these laws. One can violate
these rules and still be said to be "playing Tic-Tac-Toe."
It is the semantics of competition that is represented in these
laws. Again, the claim of the levels hypothesis is that if laws
can't be found at the physical or syntactic level to account
for observed constraints then a semantic level may be an appropriate
level at which to describe the phenomena.
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